Re: small city
Sadly the presence of a University has no bearing on the modern legal definition of a city in the UK. If it has a royal charter to make it a city, then it is a city. Otherwise, it is not.
Take Guildford - University, Cathedral, decent size. City? No - it has no royal charter, thus it is not a city. In fact, there are no cities in the whole of Surrey (a county so strange that the County Council HQ is actually outside of the county!)
Take St Asaph in sleepy North Wales. Smallest cathedral in the country, no university, population of a large village (~3-4k). City? Yes - it was granted its royal charter for the most recent jubilee, and thus it is, quite laughably, a city. The only city in Denbighshire (Denbigh is not a city) so maybe it should be renamed St Asaphshire or Llanelwyshire / Sir Llanelwy?